Note: Please use the tabs to navigate through output files

(1) Aims of our modeling:

To estimate the number of individuals hospitalized, in the ICU, on ventilators (to be added), and deaths over time in Los Angeles County as a whole and by service area (SPA)
To develop a modeling framework for scenario planning around different interventions in LA County

(2) Methods summary

(2.1) Model contributions

The incorporation of risk factors (age, sex, and other comorbidities) in the estimation of model parameter inputs

This includes in the estimation of the conditional effects for each risk factor and in the weighting of these risks by the population prevalence of combinations of these risk factors within LA County, and by SPA

Dynamic \(R0\) / infectivity rate

Looking back: Enables us to estimate the effect of social distancing interventions on spreading rate, based on observed data
Looking forward: Enables scenario planning

Direct incorporation of parameter \(r\): fraction of cases reported

Allows us to estimate this fraction directly from the model

Stochastic model:

Deterministic model: for given values of parameters, dynamics across compartments will be fixed.
Stochastic model: Appropriate probability distributions are used model the transfer of individuals across compartments.

What a stochastic model allows:

Looking back: Provides a better framework for parameter estimation, based on observed data
Looking forward: Enables forecasts with confidence bounds that account for variability in parameters

Approximate Bayesian Computation (ABC) for parameter estimation:

Allows us to incorporate the uncertainty in all the parameters in fitting the model to data and estimating parameters
Allows us to include all prior information and/or assumptions about the distribution (the range of values) for each parameter
Allows us to prioritize data input that is more reliable in fitting the model to data (e.g., not including more unreliable early illness count data)

(2.2) Model overview

Flow diagram

Compartmental model flow diagram

System of Equations

\[ \begin{align*} dS/dt &= -\beta S(I+A)\\ dE/dt &= \beta S(I+A) - \tfrac{1}{d_{EI}}E\\ dA/dt &= \tfrac{1-r}{d_{EI}}E - \tfrac{1}{d_{IR}}A\\ dI/dt &= \tfrac{r}{d_{EI}}E - (\tfrac{\alpha}{d_{IH}}\tfrac{1-\alpha}{d_{IR}})I\\ dH/dt &= \alpha (\tfrac{\alpha}{d_{IH}}\tfrac{1-\alpha}{d_{IR}})I - (\tfrac{\kappa}{d_{HQ}}\tfrac{1-\kappa}{d_{HR}})H \\ dQ/dt &= \kappa (\tfrac{\kappa}{d_{HQ}}\tfrac{1-\kappa}{d_{HR}})H - (\tfrac{\delta}{d_{QD}}\tfrac{1-\delta}{d_{QR}})Q \\ dV/dt &= p_V Q\\ dD/dt &= \delta (\tfrac{\delta}{d_{QD}}\tfrac{1-\delta}{d_{QR}})Q\\ dR/dt &= (1-\alpha) (\tfrac{\alpha}{d_{IH}}\tfrac{1-\alpha}{d_{IR}})I + (1-\kappa) (\tfrac{\kappa}{d_{HQ}}\tfrac{1-\kappa}{d_{HR}})H + (1-\delta)(\tfrac{\delta}{d_{QD}}\tfrac{1-\delta}{d_{QR}})Q + \tfrac{1}{d_{IR}}A \ \end{align*} \]

\[ R0 = \beta ({\frac{r}{\tfrac{\alpha}{d_{IH}}+\tfrac{1-\alpha}{d_{IR}}}+ (1-r){d_{IR}}}) \\ N=S+E+A+I+H+Q+D+R \]

Model parameters

Parameter	Description	Value
\(R0\)	Basic reproductive number	Estimated
\(\beta\)	transmission rate	Analytically derived from model and R0
\(d_{EI}\)	days between exposure and infectivity (incubation period)	5 days
\(d_{IH}\)	days between symptom onset and hospitalization (if required)	10 days
\(d_{IR}\)	days between symptom onset and recovery (if not hospitalized)	7 days
\(d_{HQ}\)	days between hospitalization and ICU (if required)	1 days
\(d_{QR}\)	days between hospitalization and recovery (if ICU not required)	12 days
\(d_{QD}\)	days between ICU and fatality	8 days
\(d_{QR}\)	days between ICU and recovery	7 days
\(\alpha\)	probability infected (I) requires hospitalization (vs. recovers)	Estimated
\(\kappa\)	probability hospitalized (H) requires ICU (vs. recovers)	Estimated
\(\delta\)	probability ICU (Q) patient dies	Estimated
\(p_V\)	probability ventilation (V) required given ICU	Estimated
\(N\)	Total population size
\(S\)	Susceptible population
\(E\)	Exposed not yet infectious
\(A\)	Infected, unobserved
\(I\)	Infected, observed
\(H\)	In Hospital
\(Q\)	In ICU
\(V\)	On ventilator
\(D\)	Dead
\(R\)	Recovered/removed

Model parameters - fixed, taken from literature:

Transition times between compartments
Sources provided at this link

Model Parameters — estimated through ABC (results in following section)

\(R0\), the reproductive number or average number of new infections generated by an infected person in a completely susceptible population
\(r\), the proportion of illnesses that are observed
\(Frac_{R0}\), the reduction in the initial R0 due to social distancing
\(\alpha\), the probability of hospitalization given illness
\(\kappa\), the probability of ICU care necessary given hospitalization
\(p_v\), the probability of ventilation given ICU care
\(\delta\), the probability of death given ICU care

Illness case data

We use current numbers of infected, hospitalized, ICU, ventilated, and fatal cases from Faith Washburn’s nightly reports
The data used in this version of the report (April 10th, 2020) includes counts up through April 7th, 2020

(3) Results

(3.1) Results - parameter estimates

In this section we focus on parameter estimates for key epidemic and model quantities:

\(R0\), the reproductive number or average number of new infections generated by an infected person in a completely susceptible population
\(r\), the proportion of illnesses that are observed
\(Frac_{R0}\), the reduction in the initial R0 due to social distancing
\(\alpha\), the probability of hospitalization given illness
\(\kappa\), the probability of ICU care necessary given hospitalization
\(p_v\), the probability of ventilation given ICU care
\(\delta\), the probability of death given ICU care

Summary of estimated parameters

	R0	Prop. cases observed	Start time	R0 reduction	Pr(Death)	Pr(Hospital)	Pr(ICU)	Pr(Ventilation)
mean	3.3108071	0.1841048	49.318223	0.3968724	0.5540317	0.1712917	0.2692573	0.709278
sd	0.3046965	0.0714864	7.669841	0.0653765	0.1321647	0.0308491	0.0416688	0.067001

\(R0\) Reproductive number

Mean = 3.311
Standard deviation = 0.305

## Warning: Removed 1 rows containing non-finite values (stat_density).

\(r\) proportion of observed illnesses

Mean = 0.184
Standard deviation = 0.071

\(Frac_{R0}\), the reduction in the initial R0 due to social distancing

Mean = 0.397
Standard deviation = 0.065

Information informing this parameter’s prior distribution: - Reduction in mobility observed in LA County: Source: Assessing changes in commuting and individual mobility in major metropolitan areas in the United States during the COVID-19 outbreak

Our modeled reduction in R0 timeline:

## Warning: Removed 2 row(s) containing missing values (geom_path).

\(\alpha\), the probability of hospitalization given illness

Mean = 0.171
Standard deviation = 0.031

\(\kappa\), the probability of ICU care necessary given hospitalization

Mean = 0.269
Standard deviation = 0.042

\(\delta\), the probability of death given ICU care

Mean = 0.554
Standard deviation = 0.132

\(p_v\), the probability of ventilation given ICU care

Mean = 0.709
Standard deviation = 0.067

(3.2) Results - Model forecasts

All forecasts based on stochastic model with mean value of estimated parameter distributions as input

Demonstrating model fit against data: available data

Demonstrating model fit against data, for availble variables:

Detected illnesses, cumulative = Idetectcum
Hospitalizations, cumulative = Htotcum
Deaths, cumulative = D
Ventilation, cumulative = V
Hospitalization, daily new incidence = H_new
Deaths, daily new incidence = D_new

Demonstrating model fit across all variables

Plotting model fit against data across all variables, only across available data time window

Plotting model fit against data across all variables, across full epidemic time course

Peak hospitalization, ICU, ventilation vs. capacity

Plotting total number at any point in time in Hospital, ICU, Ventilation

If everything continues as is, model projections forecast that:

The peak of the epidemic will be around the beginning of July
LA County will not come close to exceeding hospital capacity
ICU capacity may be exceeded around the peak of the epidemic

The black lines in this plot indicates capacity

(3.3) Results - Sensitivity of model projections to parameter estimates

In this section we demonstrate projections for Hospitalization (H), ICU (Q), and Ventilation (V) needs given the mean and 95% upper and lower confidence bounds around each parameter estimate

Sensitivity analysis: Mean, 95% Upper and Lower CI for \(T0\) start time

Start time = mean (49.3182228)

Start time = upper 95% CI (64.3511112)

Start time = lower 95% CI (34.2853345)

Start time = combined mean, upper, and lower 95% CI

Sensitivity analysis: Mean, 95% Upper and Lower CI for \(R0\) Reproductive number

R0 = mean (3.3108071)

R0 = upper 95% CI (3.9080122)

R0 = lower 95% CI (2.7136019)

R0 = combined mean, upper, and lower 95% CI

(4) Illustration: Model as a tool for future intervention scenario planning

Model projections forecast that, if everything continutes as is, LA is not going to see a sharp peak in illnesses in the near future, and rather will see a long, shallow, manageable epidemic curve over the next number of months.
What can we do to maintain this manageable epidemic curve while sustaining a functioning society?
This model can be used to evaluate the effect of scenarios relating to social distancing on the ultimate epidemic trajectory

Example scenarios: Relax social distancing to 25% / 0% reduction at beginning of May / June

Scenario: Relax social distancing to a 25% reduction from the norm on May 1st

## Warning: Removed 1 row(s) containing missing values (geom_path).

Scenario: Relax social distancing to 25% reduction from the norm on June 1st

## Warning: Removed 1 row(s) containing missing values (geom_path).

Scenario: Relax social distancing to 0% reduction on May 1st

## Warning: Removed 1 row(s) containing missing values (geom_path).

Scenario: Relax social distancing to 0% reduction on June 1st

## Warning: Removed 2 row(s) containing missing values (geom_path).

(5) Next steps

Integrating high-resolution near-real-time data on mobility for Los Angeles to infer contact rates
Interactive web tool
Estimating affected populations
What do you want to see here?

Acknowledgements

We would like to acknowledge Claire Jacquillat, data scientist, for her contributions related to data wrangling, cleaning, interpretation, and visualization.

Appendix

A1. Calculating Pr(H), Pr(Q), Pr(H), based on age, gender, and population prevalences of risk factors A2. Specification of stochastic model

(A1) Calculating Pr(H), Pr(Q), Pr(H), based on age, gender, and population prevalences of risk factors

Key risk profiles

Age	Sex	Smoking	Comorbidity
0-19	Female/Male	No	None
20-44	Female	No	None
45-64	Female	No	None
65+	Female	No	None
20-44	Male	No	None
45-64	Male	No	None
65+	Male	No	None
20-44	Female	Yes	None
45-64	Female	Yes	None
65+	Female	Yes	None
20-44	Male	Yes	None
45-64	Male	Yes	None
65+	Male	Yes	None
20-44	Female	No	Yes
45-64	Female	No	Yes
65+	Female	No	Yes
20-44	Male	No	Yes
45-64	Male	No	Yes
65+	Male	No	Yes
20-44	Female	Yes	Yes
45-64	Female	Yes	Yes
65+	Female	Yes	Yes
20-44	Male	Yes	Yes
45-64	Male	Yes	Yes
65+	Male	Yes	Yes

Population risk factor prevalences for LA County

Data coming from Los Angeles County Health Survey

	age.0.19	age.20.44	age.45.64	age.65	gender.Male	smoker	diabetes	hypertension	copd	coronary
Antelope Valley	0.27	0.32	0.31	0.10	0.49	0.18600	0.13100	0.30700	0.09000	0.30300
San Fernando	0.22	0.31	0.34	0.13	0.49	0.11200	0.10100	0.24200	0.07700	0.24700
San Gabriel	0.22	0.31	0.33	0.14	0.49	0.09600	0.11000	0.25500	0.05700	0.27700
Metro	0.20	0.36	0.33	0.12	0.51	0.13300	0.12100	0.25000	0.06000	0.28900
West	0.16	0.35	0.33	0.16	0.49	0.07500	0.06300	0.19600	0.05900	0.24800
South	0.29	0.34	0.28	0.08	0.49	0.12500	0.14700	0.25900	0.08500	0.26500
East	0.26	0.33	0.30	0.12	0.49	0.09300	0.11400	0.23200	0.04900	0.24900
South Bay	0.23	0.31	0.33	0.13	0.49	0.12400	0.12500	0.27600	0.06900	0.29000
LA County	0.23	0.33	0.32	0.12	0.50	0.11254	0.11315	0.24999	0.06663	0.26804

Pr(H)

Pr	age.0.19	Age20.44	Age45.64	Age65.	SexMale	SmokingYes	ComorbidityYes
0.025	0	1	0	0	0	0	0
0.103	0	1	0	0	0	0	0
0.146	0	0	1	0	0	0	0
0.204	0	0	0	1	0	0	0
0.105	0	1	0	0	1	0	0
0.149	0	0	1	0	1	0	0
0.208	0	0	0	1	1	0	0
0.173	0	1	0	0	0	1	0
0.238	0	0	1	0	0	1	0
0.319	0	0	0	1	0	1	0
0.176	0	1	0	0	1	1	0
0.242	0	0	1	0	1	1	0
0.323	0	0	0	1	1	1	0
0.161	0	1	0	0	0	0	1
0.223	0	0	1	0	0	0	1
0.300	0	0	0	1	0	0	1
0.164	0	1	0	0	1	0	1
0.227	0	0	1	0	1	0	1
0.305	0	0	0	1	1	0	1
0.259	0	1	0	0	0	1	1
0.343	0	0	1	0	0	1	1
0.439	0	0	0	1	0	1	1
0.264	0	1	0	0	1	1	1
0.348	0	0	1	0	1	1	1
0.444	0	0	0	1	1	1	1
0.025	1	0	0	0	1	0	0
0.025	1	0	0	0	0	0	0

Pr(Q)

Pr	age.0.19	Age20.44	Age45.64	Age65.	SexMale	SmokingYes	ComorbidityYes
0.000	0	1	0	0	0	0	0
0.202	0	1	0	0	0	0	0
0.259	0	0	1	0	0	0	0
0.325	0	0	0	1	0	0	0
0.241	0	1	0	0	1	0	0
0.305	0	0	1	0	1	0	0
0.377	0	0	0	1	1	0	0
0.325	0	1	0	0	0	1	0
0.399	0	0	1	0	0	1	0
0.478	0	0	0	1	0	1	0
0.377	0	1	0	0	1	1	0
0.454	0	0	1	0	1	1	0
0.535	0	0	0	1	1	1	0
0.217	0	1	0	0	0	0	1
0.277	0	0	1	0	0	0	1
0.345	0	0	0	1	0	0	1
0.259	0	1	0	0	1	0	1
0.325	0	0	1	0	1	0	1
0.399	0	0	0	1	1	0	1
0.346	0	1	0	0	0	1	1
0.421	0	0	1	0	0	1	1
0.501	0	0	0	1	0	1	1
0.399	0	1	0	0	1	1	1
0.478	0	0	1	0	1	1	1
0.558	0	0	0	1	1	1	1
0.000	1	0	0	0	1	0	0
0.000	1	0	0	0	0	0	0

Pr(D)

Pr	age.0.19	Age20.44	Age45.64	Age65.	SexMale	SmokingYes	ComorbidityYes
0.000	0	1	0	0	0	0	0
0.048	0	1	0	0	0	0	0
0.100	0	0	1	0	0	0	0
0.197	0	0	0	1	0	0	0
0.050	0	1	0	0	1	0	0
0.104	0	0	1	0	1	0	0
0.204	0	0	0	1	1	0	0
0.099	0	1	0	0	0	1	0
0.195	0	0	1	0	0	1	0
0.348	0	0	0	1	0	1	0
0.103	0	1	0	0	1	1	0
0.202	0	0	1	0	1	1	0
0.358	0	0	0	1	1	1	0
0.053	0	1	0	0	0	0	1
0.110	0	0	1	0	0	0	1
0.214	0	0	0	1	0	0	1
0.055	0	1	0	0	1	0	1
0.115	0	0	1	0	1	0	1
0.222	0	0	0	1	1	0	1
0.109	0	1	0	0	0	1	1
0.212	0	0	1	0	0	1	1
0.372	0	0	0	1	0	1	1
0.113	0	1	0	0	1	1	1
0.220	0	0	1	0	1	1	1
0.383	0	0	0	1	1	1	1
0.000	1	0	0	0	1	0	0
0.000	1	0	0	0	0	0	0

Frequency of each profile across the SPAs

Antelope Valley	San Fernando	San Gabriel	Metro	West	South	East	South Bay	LA County	age.0.19	Age20.44	Age45.64	Age65.	SexMale	SmokingYes	ComorbidityYes
0.093	0.125	0.080	0.126	0.114	0.117	0.102	0.087	0.095	0	1	0	0	0	0	0
0.093	0.125	0.080	0.126	0.114	0.117	0.102	0.087	0.095	0	1	0	0	0	0	0
0.051	0.063	0.104	0.071	0.082	0.054	0.068	0.077	0.044	0	0	1	0	0	0	0
0.007	0.023	0.028	0.011	0.028	0.014	0.012	0.019	0.012	0	0	0	1	0	0	0
0.061	0.079	0.066	0.086	0.086	0.099	0.095	0.070	0.091	0	1	0	0	1	0	0
0.071	0.049	0.064	0.073	0.072	0.031	0.080	0.053	0.047	0	0	1	0	1	0	0
0.015	0.005	0.012	0.009	0.019	0.005	0.010	0.017	0.009	0	0	0	1	1	0	0
0.024	0.007	0.014	0.004	0.002	0.005	0.005	0.017	0.005	0	1	0	0	0	1	0
0.002	0.014	0.007	0.009	0.014	0.012	0.010	0.010	0.019	0	0	1	0	0	1	0
0.000	0.002	0.002	0.007	0.000	0.000	0.000	0.002	0.000	0	0	0	1	0	1	0
0.010	0.014	0.005	0.013	0.009	0.007	0.010	0.010	0.014	0	1	0	0	1	1	0
0.015	0.012	0.007	0.007	0.012	0.005	0.007	0.012	0.012	0	0	1	0	1	1	0
0.002	0.000	0.000	0.002	0.005	0.002	0.000	0.002	0.002	0	0	0	1	1	1	0
0.093	0.049	0.066	0.042	0.054	0.073	0.063	0.062	0.072	0	1	0	0	0	0	1
0.068	0.102	0.071	0.077	0.068	0.089	0.078	0.096	0.091	0	0	1	0	0	0	1
0.022	0.021	0.038	0.033	0.030	0.021	0.029	0.038	0.033	0	0	0	1	0	0	1
0.083	0.072	0.061	0.062	0.054	0.070	0.063	0.075	0.086	0	1	0	0	1	0	1
0.093	0.081	0.106	0.077	0.086	0.063	0.075	0.087	0.102	0	0	1	0	1	0	1
0.037	0.023	0.056	0.040	0.028	0.021	0.036	0.041	0.014	0	0	0	1	1	0	1
0.017	0.007	0.005	0.007	0.007	0.016	0.012	0.002	0.019	0	1	0	0	0	1	1
0.017	0.009	0.009	0.004	0.005	0.021	0.002	0.017	0.009	0	0	1	0	0	1	1
0.007	0.005	0.005	0.004	0.014	0.005	0.005	0.007	0.002	0	0	0	1	0	1	1
0.012	0.009	0.002	0.018	0.005	0.009	0.007	0.014	0.005	0	1	0	0	1	1	1
0.020	0.014	0.007	0.018	0.009	0.016	0.007	0.014	0.016	0	0	1	0	1	1	1
0.005	0.002	0.002	0.009	0.002	0.002	0.005	0.002	0.007	0	0	0	1	1	1	1
0.041	0.035	0.054	0.024	0.040	0.054	0.061	0.046	0.053	1	0	0	0	1	0	0
0.041	0.053	0.049	0.040	0.040	0.070	0.054	0.036	0.047	1	0	0	0	0	0	0

Calculated risk probabilities

Probability of hospitalization = Pr(H)
Probability of ICU, given hospitalization = Pr(Q)

Probability of death, given ICU = Pr(D)

	Pr(H)	Pr(Q)	Pr(D)
Antelope Valley	0.162	0.243	0.088
San Fernando	0.149	0.225	0.081
San Gabriel	0.160	0.238	0.092
Metro	0.156	0.237	0.089
West	0.153	0.234	0.089
South	0.145	0.214	0.076
East	0.147	0.223	0.080
South Bay	0.165	0.246	0.094
LA County	0.155	0.233	0.083

(A2) Specification of stochastic model

r_output(readLines(path_seihqdr_model))

## ```r
## 
## # TRANSITION EQUATIONS
## 
## ## Core equations for transitions between compartments:
## update(S) <- S - n_SE
## update(E) <- E + n_SE - n_Eout
## update(I) <- I + n_EoutI - n_Iout
## update(A) <- A + n_EoutA - n_AR
## update(H) <- H + n_IoutH - n_Hout
## update(Q) <- Q + n_HoutQ - n_Qout
## update(D) <- D + n_QoutD
## update(R) <- R + n_IoutR + n_HoutR + n_QoutR + n_AR
## 
## ## Htot = H + Q
## update(Htot) <- H + Q + n_IoutH - n_HoutR - n_Qout  # Htot represents all in Hospital: Non-ICU + ICU
## 
## ## Ventilators (tracking as frac of Q, do not go to other compartments)
## update(V) <- p_QV*Q            #V + n_QV - n_Vout
## 
## ## Tracking cumulative numbers in compartments:
## update(Idetectcum) <- Idetectcum + n_EoutI
## update(Itotcum) <- Itotcum + n_Eout
## update(Htotcum) <- Htotcum + n_IoutH   #Htotcum represents cumulative of all in Hospital: Non-ICU + ICU
## update(Qcum) <- Qcum + n_HoutQ
## update(Vcum) <- p_QV*Qcum      #Vcum + n_QV
## 
## ## New daily numbers
## output(I_detect_new) <- n_EoutI
## output(I_tot_new) <- n_Eout
## output(H_new) <- n_IoutH
## output(Q_new) <- n_HoutQ
## output(D_new) <- n_QoutD
## #output(d_EI_rand) <- d_EI
## 
## ####################################################################################
## 
## # PROBABILITIES
## 
## ## Individual probabilities of transition:
## p_SE <- 1 - exp(-(Beta * (I+A)) / N)                         # S to E
## p_Eout <- 1 - exp(-1/d_EI)                               # E to I
## p_Iout <- 1 - exp(-((Alpha/d_IH) + ((1-Alpha)/d_IR)))  #exp(-((1/d_IH) + (1/d_IR)))                        # I to H and R
## p_Hout <- 1 - exp(-((Kappa/d_HQ) + ((1-Kappa)/d_HR)))  #exp(-((1/d_HQ) + (1/d_HR)))                        # H to Q and R
## p_Qout <- 1 - exp(-((Delta/d_QD) + ((1-Delta)/d_QR)))  #exp(-((1/d_QD) + (1/d_QR)))                        # Q to D and R
## p_AR <- 1 - exp(-1/d_IR)
## #p_Vout <- 1 - exp(-1/d_V)                              # Leaving V
## 
## 
## 
## # RANDOM DRAWS FOR NUMBERS CHANGING BETWEEN COMPARTMENTS 
## ## Draws from binomial and multinomial distributions for numbers changing between compartments:
## 
## ### S to E
## n_SE <- rbinom(S, p_SE)
## 
## ### E to I and A
## n_Eout <- rbinom(E, p_Eout)
## n_EoutIA[] <- rmultinom(n_Eout, p_EoutIA)
## p_EoutIA[1] <- r
## p_EoutIA[2] <- 1-r
## dim(p_EoutIA) <- 2
## dim(n_EoutIA) <- 2
## n_EoutI <- n_EoutIA[1]
## n_EoutA <- n_EoutIA[2]
## 
## ### A to R
## n_AR <- rbinom(A, p_AR)
## 
## ### I to H and R
## n_Iout <- rbinom(I, p_Iout)                                           # Total no. leaving I
## n_IoutHR[] <- rmultinom(n_Iout, p_IoutHR)                             # Divide total no. leaving I into I->H and I->R 
## p_IoutHR[1] <- Alpha #(Alpha/d_IH)/((Alpha/d_IH) + ((1-Alpha)/d_IR))         # Goes to H and R with relative rates
## p_IoutHR[2] <- 1-Alpha #((1-Alpha)/d_IR)/((Alpha/d_IH) + ((1-Alpha)/d_IR))     # 1-p_IoutHR[1]
## dim(p_IoutHR) <- 2
## dim(n_IoutHR) <- 2
## n_IoutH <- n_IoutHR[1]                                                # Total no. I->H
## n_IoutR <- n_IoutHR[2]                                                # Total no. I->R
## 
## ### H to Q and R
## n_Hout <- rbinom(H, p_Hout)
## n_HoutQR[] <- rmultinom(n_Hout, p_HoutQR)
## p_HoutQR[1] <- Kappa #(Kappa/d_HQ)/((Kappa/d_HQ) + ((1-Kappa)/d_HR)) 
## p_HoutQR[2] <- 1-Kappa #((1-Kappa)/d_HR)/((Kappa/d_HQ) + ((1-Kappa)/d_HR))
## dim(p_HoutQR) <- 2
## dim(n_HoutQR) <- 2
## n_HoutQ <- n_HoutQR[1]
## n_HoutR <- n_HoutQR[2]
## 
## ### Q to D and R
## n_Qout <- rbinom(Q, p_Qout)
## n_QoutDR[] <- rmultinom(n_Qout, p_QoutDR)
## p_QoutDR[1] <- Delta #(Delta/d_QD)/((Delta/d_QD) + ((1-Delta)/d_QR)) 
## p_QoutDR[2] <- 1-Delta #((1-Delta)/d_QR)/((Delta/d_QD) + ((1-Delta)/d_QR))
## dim(p_QoutDR) <- 2
## dim(n_QoutDR) <- 2
## n_QoutD <- n_QoutDR[1]
## n_QoutR <- n_QoutDR[2]
## 
## ### Q to V and Vout
## #n_QV <- rbinom(Q, p_QV)
## #n_Vout <- rbinom(V, p_Vout)
## 
## ######################################################################
## 
## # TOTAL POPULATION SIZE
## N <- S + E + I + A + H + Q + D + R
## 
## ######################################################################
## 
## # INITIAL STATES
## ## Core compartments
## initial(S) <- S_ini
## initial(E) <- E_ini
## initial(I) <- 0
## initial(A) <- 0
## initial(H) <- 0
## initial(Q) <- 0
## initial(D) <- 0
## initial(R) <- 0
## initial(V) <- 0
## initial(Htot) <- 0
## 
## ## Cumulative counts
## initial(Idetectcum) <- 0
## initial(Itotcum) <- 0
## initial(Htotcum) <- 0
## initial(Qcum) <- 0
## initial(Vcum) <- 0
## 
## ######################################################################
## 
## # USER DEFINED PARAMETERS 
## ## Default in parentheses:
## 
## ### Initial conditions
## S_ini <- user(1e7) # susceptibles
## E_ini <- user(10) # infected
## 
## ### Parameters - random
## #d_EI <- runif(3, 8)
## 
## ### Parameters - fixed
## d_EI <- user(5.2)  #days between exposure and infectivity (incubation period)
## d_IH <- user(10)   #days between illness onset and hospitalization
## d_IR <- user(7)    #days between illness onset and recovery (hospitalization not required)       
## d_HQ <- user(1)    #days between hospitalization start and ICU
## d_HR <- user(12)   #days in hospital (ICU not required)
## d_QD <- user(8)    #days in ICU before death (given death)
## d_QR <- user(7)    #days in ICU before recovery (given recovery)
## #d_V <- user(3)     #days on ventilator (within ICU)
## 
## ### Parameters - weighted average risk probabilities: input from JAM + population prevalence
## Alpha <- user(0.14)   #probability infected (I) requires hospitalization (vs. recovers)
## Kappa <- user(0.23)   #probability hospitalized (H) requires ICU (vs. recovers)
## Delta <- user(0.06)   #probability ICU (Q) patient dies 
## p_QV <- user(0.667)   #probability in ICU and requires ventilation
## r <- user(0.25)
## 
## ### Other variables
## #R0 <- user(2.2)     #Current estimates from other models
## 
## ### Parameters - calculated from inputs
## #Br <- R0 * ( 1 / ( (r/ ((Alpha/d_IH) + ((1-Alpha)/d_IR)))  + (1-r)*d_IR )) 
## 
## 
## 
## #########################################
## ### TIME VARYING BETA (INTERPOLATION) ###
## #########################################
## 
## Beta <- interpolate(Beta_t, Beta_y,"linear")
## 
## Beta_t[] <- user()# R0 * ((Alpha/d_IH)+((1-Alpha)/d_IR))
## Beta_y[] <- user()
## dim(Beta_t) <- user()
## dim(Beta_y) <- user()
## 
## 
## 
## ```

Predictive Epidemic Model for LA County

Abigail Horn, Lai Jiang, Wendy Cozen, David Conti, Department of Preventive Medicine, USC

4/10/2020